Halperin conjecture

In rational homotopy theory, the Halperin conjecture concerns the Serre spectral sequence of certain fibrations. It is named after the Canadian mathematician Stephen Halperin.

Statement

Suppose that ${\displaystyle F\to E\to B}$ is a fibration of simply connected spaces such that ${\displaystyle F}$ is rationally elliptic and ${\displaystyle \chi (F)\neq 0}$ (i.e., ${\displaystyle F}$ has non-zero Euler characteristic), then the Serre spectral sequence associated to the fibration collapses at the ${\displaystyle E_{2}}$ page. ^[1]

Status

As of 2019, Halperin's conjecture is still open. Gregory Lupton has reformulated the conjecture in terms of formality relations. ^[2]

Notes

1. Berglund, Alexander (2012), Rational homotopy theory (PDF)
2. Lupton, Gregory (1997), "Variations on a conjecture of Halperin", Homotopy and Geometry (Warsaw, 1997), arXiv: math/0010124 , MR   1679854

Further reading

• Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (1993), "Elliptic spaces II", L'Enseignement Mathématique , 39 (1–2): 25, doi:10.5169/seals-60412, MR   1225255
• Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2001), Rational Homotopy Theory, New York: Springer Nature, doi:10.1007/978-1-4613-0105-9, ISBN   0-387-95068-0, MR   1802847
• Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2015), Rational Homotopy Theory II, Singapore: World Scientific, doi:10.1142/9473, ISBN   978-981-4651-42-4, MR   3379890
• Félix, Yves; Oprea, John; Tanré, Daniel (2008), Algebraic Models in Geometry, Oxford: Oxford University Press, ISBN   978-0-19-920651-3, MR   2403898
• Griffiths, Phillip A.; Morgan, John W. (1981), Rational Homotopy Theory and Differential Forms, Boston: Birkhäuser, ISBN   3-7643-3041-4, MR   0641551
• Hess, Kathryn (1999), "A history of rational homotopy theory", in James, Ioan M. (ed.), History of Topology, Amsterdam: North-Holland, pp. 757–796, doi:10.1016/B978-044482375-5/50028-6, ISBN   0-444-82375-1, MR   1721122
• Hess, Kathryn (2007), "Rational homotopy theory: a brief introduction" (PDF), Interactions between Homotopy Theory and Algebra, Contemporary Mathematics, vol. 436, American Mathematical Society, pp. 175–202, arXiv: math/0604626 , doi:10.1090/conm/436/08409, ISBN   9780821838143, MR   2355774